OpenRaiser/Intern-Atlas huggingface dataset:
Constant-Rank Codes and Their Connection to Constant-Dimension Codes
Venue: arXiv cs.IT 2010
Year: 2010

Constant-dimension codes have recently received attention due to their significance to error control in noncoherent random linear network coding. What the maximal cardinality of any constant-dimension code with finite dimension and minimum distance is and how to construct the optimal constant-dimension code (or codes) that achieves the maximal cardinality both remain open research problems. In this paper, we introduce a new approach to solving these two problems. We first establish a connection between constant-rank codes and constant-dimension codes. Via this connection, we show that optimal constant-dimension codes correspond to optimal constant-rank codes over matrices with sufficiently many rows. As such, the two aforementioned problems are equivalent to determining the maximum cardinality of constant-rank codes and to constructing optimal constant-rank codes, respectively. To this end, we then derive bounds on the maximum cardinality of a constant-rank code with a given minimum rank distance, propose explicit constructions of optimal or asymptotically optimal constant-rank codes, and establish asymptotic bounds on the maximum rate of a constant-rank code.
Information theoretic bounds for Compressed Sensing
Venue: arXiv cs.IT 2010
Year: 2010

In this paper we derive information theoretic performance bounds to sensing and reconstruction of sparse phenomena from noisy projections. We consider two settings: output noise models where the noise enters after the projection and input noise models where the noise enters before the projection. We consider two types of distortion for reconstruction: support errors and mean-squared errors. Our goal is to relate the number of measurements, $m$, and $\snr$, to signal sparsity, $k$, distortion level, $d$, and signal dimension, $n$. We consider support errors in a worst-case setting. We employ different variations of Fano's inequality to derive necessary conditions on the number of measurements and $\snr$ required for exact reconstruction. To derive sufficient conditions we develop new insights on max-likelihood analysis based on a novel superposition property. In particular this property implies that small support errors are the dominant error events. Consequently, our ML analysis does not suffer the conservatism of the union bound and leads to a tighter analysis of max-likelihood. These results provide order-wise tight bounds. For output noise models we show that asymptotically an $\snr$ of $Θ(\log(n))$ together with $Θ(k \log(n/k))$ measurements is necessary and sufficient for exact support recovery. Furthermore, if a small fraction of support errors can be tolerated, a constant $\snr$ turns out to be sufficient in the linear sparsity regime. In contrast for input noise models we show that support recovery fails if the number of measurements scales as $o(n\log(n)/SNR)$ implying poor compression performance for such cases. We also consider Bayesian set-up and characterize tradeoffs between mean-squared distortion and the number of measurements using rate-distortion theory.
Alternating Automata on Data Trees and XPath Satisfiability
Venue: arXiv cs.LO 2010
Year: 2010

A data tree is an unranked ordered tree whose every node is labelled by a letter from a finite alphabet and an element ("datum") from an infinite set, where the latter can only be compared for equality. The article considers alternating automata on data trees that can move downward and rightward, and have one register for storing data. The main results are that nonemptiness over finite data trees is decidable but not primitive recursive, and that nonemptiness of safety automata is decidable but not elementary. The proofs use nondeterministic tree automata with faulty counters. Allowing upward moves, leftward moves, or two registers, each causes undecidability. As corollaries, decidability is obtained for two data-sensitive fragments of the XPath query language.
Universal Coding on Infinite Alphabets: Exponentially Decreasing Envelopes
Venue: arXiv cs.IT 2010
Year: 2010

This paper deals with the problem of universal lossless coding on a countable infinite alphabet. It focuses on some classes of sources defined by an envelope condition on the marginal distribution, namely exponentially decreasing envelope classes with exponent $α$. The minimax redundancy of exponentially decreasing envelope classes is proved to be equivalent to $\frac{1}{4 α\log e} \log^2 n$. Then a coding strategy is proposed, with a Bayes redundancy equivalent to the maximin redundancy. At last, an adaptive algorithm is provided, whose redundancy is equivalent to the minimax redundancy
Unsatisfiable CNF Formulas need many Conflicts
Venue: arXiv cs.DM 2010
Year: 2010

A pair of clauses in a CNF formula constitutes a conflict if there is a variable that occurs positively in one clause and negatively in the other. A CNF formula without any conflicts is satisfiable. The Lovasz Local Lemma implies that a k-CNF formula is satisfiable if each clause conflicts with at most 2^k/e-1 clauses. It does not, however, give any good bound on how many conflicts an unsatisfiable formula has globally. We show here that every unsatisfiable k-CNF formula requires 2.69^k conflicts and there exist unsatisfiable k-CNF formulas with 3.51^k conflicts.
Computational Approaches to Measuring the Similarity of Short Contexts : A Review of Applications and Methods
Venue: arXiv cs.CL 2010
Year: 2010

Measuring the similarity of short written contexts is a fundamental problem in Natural Language Processing. This article provides a unifying framework by which short context problems can be categorized both by their intended application and proposed solution. The goal is to show that various problems and methodologies that appear quite different on the surface are in fact very closely related. The axes by which these categorizations are made include the format of the contexts (headed versus headless), the way in which the contexts are to be measured (first-order versus second-order similarity), and the information used to represent the features in the contexts (micro versus macro views). The unifying thread that binds together many short context applications and methods is the fact that similarity decisions must be made between contexts that share few (if any) words in common.
Multitask Efficiencies in the Decision Tree Model
Venue: arXiv cs.CC 2010
Year: 2010

In Direct Sum problems [KRW], one tries to show that for a given computational model, the complexity of computing a collection of finite functions on independent inputs is approximately the sum of their individual complexities. In this paper, by contrast, we study the diversity of ways in which the joint computational complexity can behave when all the functions are evaluated on a common input. We focus on the deterministic decision tree model, with depth as the complexity measure; in this model we prove a result to the effect that the 'obvious' constraints on joint computational complexity are essentially the only ones. The proof uses an intriguing new type of cryptographic data structure called a `mystery bin' which we construct using a small polynomial separation between deterministic and unambiguous query complexity shown by Savicky. We also pose a variant of the Direct Sum Conjecture of [KRW] which, if proved for a single family of functions, could yield an analogous result for models such as the communication model.
Dynamic Connectivity in ALOHA Ad Hoc Networks
Venue: arXiv cs.IT 2010
Year: 2010

In a wireless network the set of transmitting nodes changes frequently because of the MAC scheduler and the traffic load. Previously, connectivity in wireless networks was analyzed using static geometric graphs, and as we show leads to an overly constrained design criterion. The dynamic nature of the transmitting set introduces additional randomness in a wireless system that improves the connectivity, and this additional randomness is not captured by a static connectivity graph. In this paper, we consider an ad hoc network with half-duplex radios that uses multihop routing and slotted ALOHA for the MAC contention and introduce a random dynamic multi-digraph to model its connectivity. We first provide analytical results about the degree distribution of the graph. Next, defining the path formation time as the minimum time required for a causal path to form between the source and destination on the dynamic graph, we derive the distributional properties of the connection delay using techniques from first-passage percolation and epidemic processes. We consider the giant component of the network formed when communication is noise-limited (by neglecting interference). Then, in the presence of interference, we prove that the delay scales linearly with the source-destination distance on this giant component. We also provide simulation results to support the theoretical results.
Instruction sequences and non-uniform complexity theory
Venue: arXiv cs.CC 2010
Year: 2010

We develop theory concerning non-uniform complexity in a setting in which the notion of single-pass instruction sequence considered in program algebra is the central notion. We define counterparts of the complexity classes P/poly and NP/poly and formulate a counterpart of the complexity theoretic conjecture that NP is not included in P/poly. In addition, we define a notion of completeness for the counterpart of NP/poly using a non-uniform reducibility relation and formulate complexity hypotheses which concern restrictions on the instruction sequences used for computation. We think that the theory developed opens up an additional way of investigating issues concerning non-uniform complexity.
Embedding Non-Ground Logic Programs into Autoepistemic Logic for Knowledge Base Combination
Venue: arXiv cs.LO 2010
Year: 2010

In the context of the Semantic Web, several approaches to the combination of ontologies, given in terms of theories of classical first-order logic and rule bases, have been proposed. They either cast rules into classical logic or limit the interaction between rules and ontologies. Autoepistemic logic (AEL) is an attractive formalism which allows to overcome these limitations, by serving as a uniform host language to embed ontologies and nonmonotonic logic programs into it. For the latter, so far only the propositional setting has been considered. In this paper, we present three embeddings of normal and three embeddings of disjunctive non-ground logic programs under the stable model semantics into first-order AEL. While the embeddings all correspond with respect to objective ground atoms, differences arise when considering non-atomic formulas and combinations with first-order theories. We compare the embeddings with respect to stable expansions and autoepistemic consequences, considering the embeddings by themselves, as well as combinations with classical theories. Our results reveal differences and correspondences of the embeddings and provide useful guidance in the choice of a particular embedding for knowledge combination.
Computing voting power in easy weighted voting games
Venue: arXiv cs.GT 2010
Year: 2010

Weighted voting games are ubiquitous mathematical models which are used in economics, political science, neuroscience, threshold logic, reliability theory and distributed systems. They model situations where agents with variable voting weight vote in favour of or against a decision. A coalition of agents is winning if and only if the sum of weights of the coalition exceeds or equals a specified quota. The Banzhaf index is a measure of voting power of an agent in a weighted voting game. It depends on the number of coalitions in which the agent is the difference in the coalition winning or losing. It is well known that computing Banzhaf indices in a weighted voting game is NP-hard. We give a comprehensive classification of weighted voting games which can be solved in polynomial time. Among other results, we provide a polynomial ($O(k{(\frac{n}{k})}^k)$) algorithm to compute the Banzhaf indices in weighted voting games in which the number of weight values is bounded by $k$.
Dilation, smoothed distance, and minimization diagrams of convex functions
Venue: arXiv cs.CG 2010
Year: 2010

We study Voronoi diagrams for distance functions that add together two convex functions, each taking as its argument the difference between Cartesian coordinates of two planar points. When the functions do not grow too quickly, then the Voronoi diagram has linear complexity and can be constructed in near-linear randomized expected time. Additionally, the level sets of the distances from the sites form a family of pseudocircles in the plane, all cells in the Voronoi diagram are connected, and the set of bisectors separating any one cell in the diagram from each of the others forms an arrangement of pseudolines in the plane. We apply these results to the smoothed distance or biotope transform metric, a geometric analogue of the Jaccard distance whose Voronoi diagrams can be used to determine the dilation of a star network with a given hub. For sufficiently closely spaced points in the plane, the Voronoi diagram of smoothed distance has linear complexity and can be computed efficiently. We also experiment with a variant of Lloyd's algorithm, adapted to smoothed distance, to find uniformly spaced point samples with exponentially decreasing density around a given point.
Client-server multi-task learning from distributed datasets
Venue: arXiv cs.LG 2010
Year: 2010

A client-server architecture to simultaneously solve multiple learning tasks from distributed datasets is described. In such architecture, each client is associated with an individual learning task and the associated dataset of examples. The goal of the architecture is to perform information fusion from multiple datasets while preserving privacy of individual data. The role of the server is to collect data in real-time from the clients and codify the information in a common database. The information coded in this database can be used by all the clients to solve their individual learning task, so that each client can exploit the informative content of all the datasets without actually having access to private data of others. The proposed algorithmic framework, based on regularization theory and kernel methods, uses a suitable class of mixed effect kernels. The new method is illustrated through a simulated music recommendation system.
Correlated Equilibria in Continuous Games: Characterization and Computation
Venue: arXiv cs.GT 2010
Year: 2010

We present several new characterizations of correlated equilibria in games with continuous utility functions. These have the advantage of being more computationally and analytically tractable than the standard definition in terms of departure functions. We use these characterizations to construct effective algorithms for approximating a single correlated equilibrium or the entire set of correlated equilibria of a game with polynomial utility functions.
An Improved Approximation Algorithm for the Column Subset Selection Problem
Venue: arXiv cs.DS 2010
Year: 2010

We consider the problem of selecting the best subset of exactly $k$ columns from an $m \times n$ matrix $A$. We present and analyze a novel two-stage algorithm that runs in $O(\min\{mn^2,m^2n\})$ time and returns as output an $m \times k$ matrix $C$ consisting of exactly $k$ columns of $A$. In the first (randomized) stage, the algorithm randomly selects $Θ(k \log k)$ columns according to a judiciously-chosen probability distribution that depends on information in the top-$k$ right singular subspace of $A$. In the second (deterministic) stage, the algorithm applies a deterministic column-selection procedure to select and return exactly $k$ columns from the set of columns selected in the first stage. Let $C$ be the $m \times k$ matrix containing those $k$ columns, let $P_C$ denote the projection matrix onto the span of those columns, and let $A_k$ denote the best rank-$k$ approximation to the matrix $A$. Then, we prove that, with probability at least 0.8, $$ \FNorm{A - P_CA} \leq Θ(k \log^{1/2} k) \FNorm{A-A_k}. $$ This Frobenius norm bound is only a factor of $\sqrt{k \log k}$ worse than the best previously existing existential result and is roughly $O(\sqrt{k!})$ better than the best previous algorithmic result for the Frobenius norm version of this Column Subset Selection Problem (CSSP). We also prove that, with probability at least 0.8, $$ \TNorm{A - P_CA} \leq Θ(k \log^{1/2} k)\TNorm{A-A_k} + Θ(k^{3/4}\log^{1/4}k)\FNorm{A-A_k}. $$ This spectral norm bound is not directly comparable to the best previously existing bounds for the spectral norm version of this CSSP. Our bound depends on $\FNorm{A-A_k}$, whereas previous results depend on $\sqrt{n-k}\TNorm{A-A_k}$; if these two quantities are comparable, then our bound is asymptotically worse by a $(k \log k)^{1/4}$ factor.
Novel Architectures and Algorithms for Delay Reduction in Back-pressure Scheduling and Routing
Venue: arXiv cs.NI 2010
Year: 2010

The back-pressure algorithm is a well-known throughput-optimal algorithm. However, its delay performance may be quite poor even when the traffic load is not close to network capacity due to the following two reasons. First, each node has to maintain a separate queue for each commodity in the network, and only one queue is served at a time. Second, the back-pressure routing algorithm may route some packets along very long routes. In this paper, we present solutions to address both of the above issues, and hence, improve the delay performance of the back-pressure algorithm. One of the suggested solutions also decreases the complexity of the queueing data structures to be maintained at each node.
Pilot Contamination and Precoding in Multi-Cell TDD Systems
Venue: arXiv cs.IT 2010
Year: 2010

This paper considers a multi-cell multiple antenna system with precoding used at the base stations for downlink transmission. For precoding at the base stations, channel state information (CSI) is essential at the base stations. A popular technique for obtaining this CSI in time division duplex (TDD) systems is uplink training by utilizing the reciprocity of the wireless medium. This paper mathematically characterizes the impact that uplink training has on the performance of such multi-cell multiple antenna systems. When non-orthogonal training sequences are used for uplink training, the paper shows that the precoding matrix used by the base station in one cell becomes corrupted by the channel between that base station and the users in other cells in an undesirable manner. This paper analyzes this fundamental problem of pilot contamination in multi-cell systems. Furthermore, it develops a new multi-cell MMSE-based precoding method that mitigate this problem. In addition to being a linear precoding method, this precoding method has a simple closed-form expression that results from an intuitive optimization problem formulation. Numerical results show significant performance gains compared to certain popular single-cell precoding methods.
Fundamental delay bounds in peer-to-peer chunk-based real-time streaming systems
Venue: arXiv cs.PF 2010
Year: 2010

This paper addresses the following foundational question: what is the maximum theoretical delay performance achievable by an overlay peer-to-peer streaming system where the streamed content is subdivided into chunks? As shown in this paper, when posed for chunk-based systems, and as a consequence of the store-and-forward way in which chunks are delivered across the network, this question has a fundamentally different answer with respect to the case of systems where the streamed content is distributed through one or more flows (sub-streams). To circumvent the complexity emerging when directly dealing with delay, we express performance in term of a convenient metric, called "stream diffusion metric". We show that it is directly related to the end-to-end minimum delay achievable in a P2P streaming network. In a homogeneous scenario, we derive a performance bound for such metric, and we show how this bound relates to two fundamental parameters: the upload bandwidth available at each node, and the number of neighbors a node may deliver chunks to. In this bound, k-step Fibonacci sequences do emerge, and appear to set the fundamental laws that characterize the optimal operation of chunk-based systems.
Fast Evaluation of Interlace Polynomials on Graphs of Bounded Treewidth
Venue: arXiv cs.DS 2010
Year: 2010

We consider the multivariate interlace polynomial introduced by Courcelle (2008), which generalizes several interlace polynomials defined by Arratia, Bollobas, and Sorkin (2004) and by Aigner and van der Holst (2004). We present an algorithm to evaluate the multivariate interlace polynomial of a graph with n vertices given a tree decomposition of the graph of width k. The best previously known result (Courcelle 2008) employs a general logical framework and leads to an algorithm with running time f(k)*n, where f(k) is doubly exponential in k. Analyzing the GF(2)-rank of adjacency matrices in the context of tree decompositions, we give a faster and more direct algorithm. Our algorithm uses 2^{3k^2+O(k)}*n arithmetic operations and can be efficiently implemented in parallel.
Languages recognized by nondeterministic quantum finite automata
Venue: arXiv cs.CC 2010
Year: 2010

The nondeterministic quantum finite automaton (NQFA) is the only known case where a one-way quantum finite automaton (QFA) model has been shown to be strictly superior in terms of language recognition power to its probabilistic counterpart. We give a characterization of the class of languages recognized by NQFA's, demonstrating that it is equal to the class of exclusive stochastic languages. We also characterize the class of languages that are recognized necessarily by two-sided error by QFA's. It is shown that these classes remain the same when the QFA's used in their definitions are replaced by several different model variants that have appeared in the literature. We prove several closure properties of the related classes. The ramifications of these results about classical and quantum sublogarithmic space complexity classes are examined.
Communication-optimal Parallel and Sequential Cholesky Decomposition
Venue: arXiv cs.NA 2010
Year: 2010

Numerical algorithms have two kinds of costs: arithmetic and communication, by which we mean either moving data between levels of a memory hierarchy (in the sequential case) or over a network connecting processors (in the parallel case). Communication costs often dominate arithmetic costs, so it is of interest to design algorithms minimizing communication. In this paper we first extend known lower bounds on the communication cost (both for bandwidth and for latency) of conventional (O(n^3)) matrix multiplication to Cholesky factorization, which is used for solving dense symmetric positive definite linear systems. Second, we compare the costs of various Cholesky decomposition implementations to these lower bounds and identify the algorithms and data structures that attain them. In the sequential case, we consider both the two-level and hierarchical memory models. Combined with prior results in [13, 14, 15], this gives a set of communication-optimal algorithms for O(n^3) implementations of the three basic factorizations of dense linear algebra: LU with pivoting, QR and Cholesky. But it goes beyond this prior work on sequential LU by optimizing communication for any number of levels of memory hierarchy.